How Traders Manage Their Exposures 1 A Trader’s Gold Portfolio. How Should Risks Be Hedged? Position Spot Gold Forward Contracts Futures Contracts Swaps Options Exotics Total Value ($) 180,000 – 60,000 2,000 80,000 –110,000 25,000 117,000 2 Delta • Delta of a portfolio is the partial derivative of a portfolio value with respect to the price of the underlying asset (gold in this case) Delta = ¶Õ ¶S • Suppose that a $0.1 increase in the price of gold leads to the gold portfolio decreasing in value by $100 • The delta of the portfolio is -1000 - 100 Delta = 0.1 3 delta neutral • The portfolio could be hedged against short-term changes in the price of gold by buying 1000 ounces of gold. This is known as making the portfolio delta neutral • Delta neutral portfolio has zero delta. 4 Linear vs Nonlinear Products • When the price of a product is linearly dependent on the price of an underlying asset, (e.g., forward contracts, futures contracts and swaps) a ``hedge and forget’’ strategy can be used (since delta does not change at all) • Non-linear products require the hedge to be rebalanced to preserve delta neutrality 5 Example • A bank has sold for 100,000 European call option on 100,000 shares of a nondividend paying stock for $300,000. • S0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks, µ = 13% • The Black-Scholes value of the option is $240,000 • How does the bank hedge its risk to lock in a $60,000 profit? 6 Delta of the Option Option price Slope = D B A Stock price 7 Delta Hedging • • • • Initially the delta of the option is 0.522 The delta of the portfolio is -0.522*100,000 =-52,200 This means that 52,200 shares are purchased to create a delta neutral position • But, if a week later delta of the option changes to 0.458, then the portfolio delta changes to -45,800 and 6,400 shares must be sold to maintain delta neutrality 8 Option closes in the money Week Stock Price Delta Shares Purchased 0 49.00 0.522 52,200 1 48.12 0.458 (6,400) 2 47.37 0.400 (5,800) 3 50.25 0.596 19,600 …. ….. …. ….. 19 55.87 1.000 1,000 20 57.25 1.000 0 Gamma • Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset ¶ 2P G= ¶S 2 • If Gamma is large in absolute term, then delta is highly sensitive to the price of the underlying asset 10 Gamma Measures the Delta Hedging Errors Caused By Curvature Call price C'' C' C Stock price S S' 11 Gamma Neutral Portfolio • A linear product has zero gamma. • A delta neutral portfolio has a gamma equal to G • To make this portfolio gamma neutral, the numberwT of traded option added to the portfolio solves wT GT + G = 0 • The Delta needs to be rebalanced 12 Vega • Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility ¶P n= ¶s • If the vega is high in absolute term, the portfolio value is very sensitive to small changes in volatility. 13 • To make this portfolio vega neutral, the number of traded option added to the portfolio solves wTn T +n = 0 14 Example Consider a portfolio that is delta neutral, with a gamma of -5000 and a vega of -8000. A traded option has a gamma of 0.5, a vega of 2.0, and a delta of 0.6. (1) How do you make the portfolio vega neutral? As a result, what are the new delta and gamma for the new portfolio? 15 (2) Suppose that there is a second traded option with a gamma of 0.8, a vega of 1.2, and a delta of 0.5. How do you make the portfolio both gamma and vega neutral? What is the resulting new delta? How to make the final portfolio delta netrual as well? 16 Solution (1) -8000+w(2)=0, w=4000 (long 4000 traded options) As a result, the new delta is 0.6*4000=2400 (sell or short 2400 units of the assets to maintain delta neutrality) The new gamma is -5000+0.5*4000=-3000 17 - 5,000 + 0.5w1 + 0.8w2 = 0 - 8,000 + 2.0w1 + 1.2 w2 = 0 w1 = 400 w2 = 6,000 400 ´ 0.6 + 6,000 ´ 0.5 = 3,240. 18 The portfolio can therefore be made gamma and vega neutral by including 400 of the first traded option and 6000 of the second traded option. The delta of the portfolio after the addition of the positions in the two traded options is 3240 Hence, 3240 units of the asset would have to be sold to maintain delta neutrality. 19 Theta • Theta (Q) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time • The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of the option declines 20 Rho • Rho is the partial derivative with respect to to a parallel shift in all interest rates in a particular country 21 Taylor Series Expansion (assume volatility and interest rates are constant) ¶P ¶P 1¶ P 2 DP = DS + Dt + (DS ) + 2 ¶S ¶t 2 ¶S 2 2 1¶ P ¶ P 2 (Dt ) + DSDt + ! 2 2 ¶t ¶t¶S 2 22 ¶P ¶P 1¶ P 2 DP = DS + Dt + (DS ) + 2 ¶S ¶t 2 ¶S 2 2 1¶ P ¶ P 2 (Dt ) + DSDt + ! 2 2 ¶t ¶t¶S 2 23 24 Interpretation of Gamma • For a delta neutral portfolio, D » Q Dt + ½GDS 2 D D DS DS Positive Gamma Negative Gamma Taylor Series Expansion when Volatility is change but interest rates are unchanged ¶P ¶P ¶P 1¶ P 2 DP = DS + Ds + Dt + ( D S ) 2 ¶S ¶s ¶t 2 ¶S 1 ¶ 2P 2 + (Ds) + 2 2 ¶s 2 26 Managing Delta, Gamma, & Vega • D can be changed by taking a position in the underlying • To adjust G & n it is necessary to take a position in an option or other derivative 27 Hedging in Practice • • • Traders usually ensure that their portfolios are delta-neutral at least once a day Whenever the opportunity arises, they improve gamma and vega A trader responsible for all trading involving a particular asset must keep gamma and vega within limits set by risk management 28